It works for me when I print $string:

my $cell_name = "xyz";
while (<DATA>) {
    if (/STARTING_PATTERN\s+(.*)/) {
        $string = $_;
        $sub    = quotemeta $1;
        $string =~ s/$sub/$cell_name/;
        print $string;
    }
}

__DATA__
STARTING_PATTERN RRR
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Output:

STARTING_PATTERN xyz
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Perhaps your input isn't as simple as RRR. If it contains regex metacharacters, use quotemeta, as shown.